Let {X, X_n; n ≥ 1} be a sequence of real-valued i.i.d. random variables and let S_n = ∑_(i=1)~n X_i, n ≥ 1. In this paper, we study the probabilities of large deviations of the form P (S_n > tn~(1/p)), P(S_n < -tn~(1/p)), and P(|S_n| > tn~(1/p)), where t > 0 and 0 < p < 2. We obtain precise asymptotic estimates for these probabilities under mild and easily verifiable conditions. For example, we show that if S_n~(1/p) → P~0 and if there exists a nonincreasing positive function φ(x) on [0,∞) which is regularly varying with index α ≤ -1 such that lim sup x → ∞ P(|X| > x~(1/p))/φ(x) = 1, then for every t > 0, lim sup n → ∞ P(|S_n| > tn~(1/p))/(nφ(n)) = t~(pα).
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